Which of these two numerical methods for z-score normalisation is preferable, in multivariate linear regression?
No, there is no computational advantage of the second method over the first, if you neglect the computational requirements for the calculation of $\sigma$ and $\mu$.

We generally use the first method for better results. This is because if you separate your dataset into train and test data, then you may normalise the train data perfectly between $0$ and $1$ by taking $max$ and $min$ from the train dataset. Your algorithm will only see data between $0-1$, but there is no way to ensure normalising the test data on the same $max$ and $min$ will result in the data being between $0$ and $1$..

So a better way is to use the Gaussian Distribution method, as the only assumption it makes is the data distribution is Gaussian which is just about true for any phenomenon (after some functional transformations which might be handled by the ML algorithm). I answered a similar question here.